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Theorem isunit 18422
Description: Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
Hypotheses
Ref Expression
unit.1 𝑈 = (Unit‘𝑅)
unit.2 1 = (1r𝑅)
unit.3 = (∥r𝑅)
unit.4 𝑆 = (oppr𝑅)
unit.5 𝐸 = (∥r𝑆)
Assertion
Ref Expression
isunit (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))

Proof of Theorem isunit
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6111 . . . 4 (𝑋 ∈ (Unit‘𝑅) → 𝑅 ∈ dom Unit)
2 unit.1 . . . 4 𝑈 = (Unit‘𝑅)
31, 2eleq2s 2701 . . 3 (𝑋𝑈𝑅 ∈ dom Unit)
43elexd 3182 . 2 (𝑋𝑈𝑅 ∈ V)
5 df-br 4574 . . . 4 (𝑋 1 ↔ ⟨𝑋, 1 ⟩ ∈ )
6 elfvdm 6111 . . . . . 6 (⟨𝑋, 1 ⟩ ∈ (∥r𝑅) → 𝑅 ∈ dom ∥r)
7 unit.3 . . . . . 6 = (∥r𝑅)
86, 7eleq2s 2701 . . . . 5 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ dom ∥r)
98elexd 3182 . . . 4 (⟨𝑋, 1 ⟩ ∈ 𝑅 ∈ V)
105, 9sylbi 205 . . 3 (𝑋 1𝑅 ∈ V)
1110adantr 479 . 2 ((𝑋 1𝑋𝐸 1 ) → 𝑅 ∈ V)
12 fveq2 6084 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
1312, 7syl6eqr 2657 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r𝑟) = )
14 fveq2 6084 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
15 unit.4 . . . . . . . . . . . 12 𝑆 = (oppr𝑅)
1614, 15syl6eqr 2657 . . . . . . . . . . 11 (𝑟 = 𝑅 → (oppr𝑟) = 𝑆)
1716fveq2d 6088 . . . . . . . . . 10 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = (∥r𝑆))
18 unit.5 . . . . . . . . . 10 𝐸 = (∥r𝑆)
1917, 18syl6eqr 2657 . . . . . . . . 9 (𝑟 = 𝑅 → (∥r‘(oppr𝑟)) = 𝐸)
2013, 19ineq12d 3772 . . . . . . . 8 (𝑟 = 𝑅 → ((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
2120cnveqd 5204 . . . . . . 7 (𝑟 = 𝑅((∥r𝑟) ∩ (∥r‘(oppr𝑟))) = ( 𝐸))
22 fveq2 6084 . . . . . . . . 9 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
23 unit.2 . . . . . . . . 9 1 = (1r𝑅)
2422, 23syl6eqr 2657 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2524sneqd 4132 . . . . . . 7 (𝑟 = 𝑅 → {(1r𝑟)} = { 1 })
2621, 25imaeq12d 5369 . . . . . 6 (𝑟 = 𝑅 → (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}) = (( 𝐸) “ { 1 }))
27 df-unit 18407 . . . . . 6 Unit = (𝑟 ∈ V ↦ (((∥r𝑟) ∩ (∥r‘(oppr𝑟))) “ {(1r𝑟)}))
28 fvex 6094 . . . . . . . . . 10 (∥r𝑅) ∈ V
297, 28eqeltri 2679 . . . . . . . . 9 ∈ V
3029inex1 4718 . . . . . . . 8 ( 𝐸) ∈ V
3130cnvex 6979 . . . . . . 7 ( 𝐸) ∈ V
3231imaex 6969 . . . . . 6 (( 𝐸) “ { 1 }) ∈ V
3326, 27, 32fvmpt 6172 . . . . 5 (𝑅 ∈ V → (Unit‘𝑅) = (( 𝐸) “ { 1 }))
342, 33syl5eq 2651 . . . 4 (𝑅 ∈ V → 𝑈 = (( 𝐸) “ { 1 }))
3534eleq2d 2668 . . 3 (𝑅 ∈ V → (𝑋𝑈𝑋 ∈ (( 𝐸) “ { 1 })))
36 inss1 3790 . . . . . 6 ( 𝐸) ⊆
377reldvdsr 18409 . . . . . 6 Rel
38 relss 5115 . . . . . 6 (( 𝐸) ⊆ → (Rel → Rel ( 𝐸)))
3936, 37, 38mp2 9 . . . . 5 Rel ( 𝐸)
40 eliniseg2 5407 . . . . 5 (Rel ( 𝐸) → (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 ))
4139, 40ax-mp 5 . . . 4 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ 𝑋( 𝐸) 1 )
42 brin 4624 . . . 4 (𝑋( 𝐸) 1 ↔ (𝑋 1𝑋𝐸 1 ))
4341, 42bitri 262 . . 3 (𝑋 ∈ (( 𝐸) “ { 1 }) ↔ (𝑋 1𝑋𝐸 1 ))
4435, 43syl6bb 274 . 2 (𝑅 ∈ V → (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 )))
454, 11, 44pm5.21nii 366 1 (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cin 3534  wss 3535  {csn 4120  cop 4126   class class class wbr 4573  ccnv 5023  dom cdm 5024  cima 5027  Rel wrel 5029  cfv 5786  1rcur 18266  opprcoppr 18387  rcdsr 18403  Unitcui 18404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fv 5794  df-dvdsr 18406  df-unit 18407
This theorem is referenced by:  1unit  18423  unitcl  18424  opprunit  18426  crngunit  18427  unitmulcl  18429  unitgrp  18432  unitnegcl  18446  unitpropd  18462  isdrng2  18522  subrguss  18560  subrgunit  18563  fidomndrng  19070  invrvald  20239  elrhmunit  28953
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